Heyde theorem on locally compact Abelian groups with the connected component of zero of dimension 1

Let $X$ be a locally compact Abelian group with the connected component of zero of dimension 1. Let ${\xi }_1$ and ${\xi }_2$ be independent random variables with values in $X$ with nonvanishing characteristic functions. We prove that if a topological automorphism $\alpha$ of the group $X$ satisfies the condition $\mathrm{K}\mathrm{e}\mathrm{r}(I+\alpha )=\{0\}$ and the conditional distribution of the linear form $L_2={\xi }_1+\alpha {\xi }_2$ given $L_1={\xi }_1+{\xi }_2$ is symmetric, then the distributions of ${\xi }_j$ are convolutions of Gaussian distributions on $X$ and distributions supported in the subgroup $\{x\in X:2x=0\}$. This result can be viewed as a generalization of the well-known Heyde theorem on the characterization of the Gaussian distribution on the real line.